On the Distribution of the non-trivial Roots of Riemann's Zeta Function. An alternative Approach

TL;DR

This paper explores the distribution of non-trivial roots of Riemann's zeta function using an alternative approach involving Weierstrass products and Stieltjes integrals, providing an approximate formula for their distribution.

Contribution

It introduces a novel method to analyze the roots of Riemann's zeta function through infinite products and integral representations, offering new insights into their distribution.

Findings

Derived an approximate formula for root distribution

Compared asymptotic behaviors of infinite products and Xi(s)

Proposed an alternative analytical approach

Abstract

Expressing Weierstrass type infinite products in terms of Stieltjes integrals is discussed. The asymptotic behavior of particular types of infinite products is compared against the asymptotic behavior of the entire function Xi(s), well-known in Riemann zeta function theory. An approximate formula for the distribution of the non-trivial roots of Riemann's zeta function is obtained.